Diagonal

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In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology.

Contents

1 Polygons
2 Matrices
3 Geometry
4 See also

Polygons

As applied to a polygon, a diagonal is a line segment joining two vertices that are not adjacent. Therefore a quadrilateral has two diagonals, joining opposite pairs of vertices. For a convex polygon the diagonals run inside the polygon. This is not so for re-entrant polygons. In fact a polygon is convex if and only if the diagonals are internal.

When n is the number of vertices in a polygon and d is the number of possible different diagonals, each vertex has possible diagonals to all other vertices save for itself and the two adjacent vertices, or n-3 diagonals; this multiplied by the number of vertices is

(n − 3) × n,

which counts each diagonal twice (once for each vertex) — therefore,

[d= \frac.\, ]

Matrices

In the case of a square matrix, the main or principal diagonal is the diagonal line of entries running north-west to south-east. For example the identity matrix can be described as having entries 1 on main diagonal, and 0 elsewhere. The north-east to south-west diagonal is sometimes described as the minor diagonal. A superdiagonal entry would be one that is above, and to the right of, the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero.

Geometry

By analogy, the subset of the Cartesian product X×X of any set X with itself, consisting of all pairs (x,x), is called the diagonal. It is the graph of the identity relation. It plays an important part in geometry: for example the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal.

Quite a major role is played in geometric studies by the idea of intersecting the diagonal with itself: not directly, but by perturbing it within an equivalence class. This is related at quite a deep level with the Euler characteristic and the zeroes of vector fields. For example the circle S¹ has Betti numbers 1, 1, 0, 0, 0, ... and so Euler characteristic 0. A geometric way of saying that is to look at the diagonal on the two-torus S¹xS¹; and to observe that it can move off itself by the small motion (θ, θ) to (θ, θ + ε). In general, the intersection number of the graph of a function with the diagonal may be computed using homology via the Lefschetz fixed point theorem; the self-intersection of the diagonal is the special case of the identity function.

Diagonal

Polygons

Matrices

Geometry

See also